Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n
The harmonic number H n {\displaystyle H_{n}} with n = ⌊ x ⌋ {\displaystyle n=\lfloor x\rfloor } (red line) with its asymptotic limit γ + ln ( x ) {\displaystyle \gamma +\ln(x)} (blue line) where γ {\displaystyle \gamma } is the Euler–Mascheroni constant . In mathematics , the n -th harmonic number is the sum of the reciprocals of the first n natural numbers :
H n = 1 + 1 2 + 1 3 + ⋯ + 1 n = ∑ k = 1 n 1 k . {\displaystyle H_{n}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}}.} Starting from n = 1 , the sequence of harmonic numbers begins:
1 , 3 2 , 11 6 , 25 12 , 137 60 , … {\displaystyle 1,{\frac {3}{2}},{\frac {11}{6}},{\frac {25}{12}},{\frac {137}{60}},\dots } Harmonic numbers are related to the harmonic mean in that the n -th harmonic number is also n times the reciprocal of the harmonic mean of the first n positive integers.
Harmonic numbers have been studied since antiquity and are important in various branches of number theory . They are sometimes loosely termed harmonic series , are closely related to the Riemann zeta function , and appear in the expressions of various special functions .
The harmonic numbers roughly approximate the natural logarithm function [1] : 143 and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers . His work was extended into the complex plane by Bernhard Riemann in 1859, leading directly to the celebrated Riemann hypothesis about the distribution of prime numbers .
When the value of a large quantity of items has a Zipf's law distribution, the total value of the n most-valuable items is proportional to the n -th harmonic number. This leads to a variety of surprising conclusions regarding the long tail and the theory of network value .
The Bertrand-Chebyshev theorem implies that, except for the case n = 1 , the harmonic numbers are never integers.[2]
The first 40 harmonic numbers n Harmonic number, Hn expressed as a fraction decimal relative size 1 1 1 1
2 3 /2 1.5 1.5
3 11 /6 ~1.83333 1.83333
4 25 /12 ~2.08333 2.08333
5 137 /60 ~2.28333 2.28333
6 49 /20 2.45 2.45
7 363 /140 ~2.59286 2.59286
8 761 /280 ~2.71786 2.71786
9 7 129 /2 520 ~2.82897 2.82897
10 7 381 /2 520 ~2.92897 2.92897
11 83 711 /27 720 ~3.01988 3.01988
12 86 021 /27 720 ~3.10321 3.10321
13 1 145 993 /360 360 ~3.18013 3.18013
14 1 171 733 /360 360 ~3.25156 3.25156
15 1 195 757 /360 360 ~3.31823 3.31823
16 2 436 559 /720 720 ~3.38073 3.38073
17 42 142 223 /12 252 240 ~3.43955 3.43955
18 14 274 301 /4 084 080 ~3.49511 3.49511
19 275 295 799 /77 597 520 ~3.54774 3.54774
20 55 835 135 /15 519 504 ~3.59774 3.59774
21 18 858 053 /5 173 168 ~3.64536 3.64536
22 19 093 197 /5 173 168 ~3.69081 3.69081
23 444 316 699 /118 982 864 ~3.73429 3.73429
24 1 347 822 955 /356 948 592 ~3.77596 3.77596
25 34 052 522 467 /8 923 714 800 ~3.81596 3.81596
26 34 395 742 267 /8 923 714 800 ~3.85442 3.85442
27 312 536 252 003 /80 313 433 200 ~3.89146 3.89146
28 315 404 588 903 /80 313 433 200 ~3.92717 3.92717
29 9 227 046 511 387 /2 329 089 562 800 ~3.96165 3.96165
30 9 304 682 830 147 /2 329 089 562 800 ~3.99499 3.99499
31 290 774 257 297 357 /72 201 776 446 800 ~4.02725 4.02725
32 586 061 125 622 639 /144 403 552 893 600 ~4.05850 4.0585
33 53 676 090 078 349 /13 127 595 717 600 ~4.08880 4.0888
34 54 062 195 834 749 /13 127 595 717 600 ~4.11821 4.11821
35 54 437 269 998 109 /13 127 595 717 600 ~4.14678 4.14678
36 54 801 925 434 709 /13 127 595 717 600 ~4.17456 4.17456
37 2 040 798 836 801 833 /485 721 041 551 200 ~4.20159 4.20159
38 2 053 580 969 474 233 /485 721 041 551 200 ~4.22790 4.2279
39 2 066 035 355 155 033 /485 721 041 551 200 ~4.25354 4.25354
40 2 078 178 381 193 813 /485 721 041 551 200 ~4.27854 4.27854
Identities involving harmonic numbers [ edit ] By definition, the harmonic numbers satisfy the recurrence relation
H n + 1 = H n + 1 n + 1 . {\displaystyle H_{n+1}=H_{n}+{\frac {1}{n+1}}.} The harmonic numbers are connected to the Stirling numbers of the first kind by the relation
H n = 1 n ! [ n + 1 2 ] . {\displaystyle H_{n}={\frac {1}{n!}}\left[{n+1 \atop 2}\right].} The harmonic numbers satisfy the series identities
∑ k = 1 n H k = ( n + 1 ) H n − n {\displaystyle \sum _{k=1}^{n}H_{k}=(n+1)H_{n}-n} and
∑ k = 1 n H k 2 = ( n + 1 ) H n 2 − ( 2 n + 1 ) H n + 2 n . {\displaystyle \sum _{k=1}^{n}H_{k}^{2}=(n+1)H_{n}^{2}-(2n+1)H_{n}+2n.} These two results are closely analogous to the corresponding integral results
∫ 0 x log y d y = x log x − x {\displaystyle \int _{0}^{x}\log y\ dy=x\log x-x} and
∫ 0 x ( log y ) 2 d y = x ( log x ) 2 − 2 x log x + 2 x . {\displaystyle \int _{0}^{x}(\log y)^{2}\ dy=x(\log x)^{2}-2x\log x+2x.} Identities involving π [ edit ] There are several infinite summations involving harmonic numbers and powers of π :[3] [better source needed ]
∑ n = 1 ∞ H n n ⋅ 2 n = π 2 12 ∑ n = 1 ∞ H n 2 ( n + 1 ) 2 = 11 360 π 4 ∑ n = 1 ∞ H n 2 ( n + 1 ) 2 = 11 360 π 4 ∑ n = 1 ∞ H n n 3 = π 4 72 {\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }{\frac {H_{n}}{n\cdot 2^{n}}}&={\frac {\pi ^{2}}{12}}\\\sum _{n=1}^{\infty }{\frac {H_{n}^{2}}{(n+1)^{2}}}&={\frac {11}{360}}\pi ^{4}\\\sum _{n=1}^{\infty }{\frac {H_{n}^{2}}{(n+1)^{2}}}&={\frac {11}{360}}\pi ^{4}\\\sum _{n=1}^{\infty }{\frac {H_{n}}{n^{3}}}&={\frac {\pi ^{4}}{72}}\end{aligned}}} Calculation [ edit ] An integral representation given by Euler [4] is
H n = ∫ 0 1 1 − x n 1 − x d x . {\displaystyle H_{n}=\int _{0}^{1}{\frac {1-x^{n}}{1-x}}\,dx.} The equality above is straightforward by the simple algebraic identity
1 − x n 1 − x = 1 + x + ⋯ + x n − 1 . {\displaystyle {\frac {1-x^{n}}{1-x}}=1+x+\cdots +x^{n-1}.} Using the substitution x = 1 − u , another expression for H n is
H n = ∫ 0 1 1 − x n 1 − x d x = ∫ 0 1 1 − ( 1 − u ) n u d u = ∫ 0 1 [ ∑ k = 1 n ( n k ) ( − u ) k − 1 ] d u = ∑ k = 1 n ( n k ) ∫ 0 1 ( − u ) k − 1 d u = ∑ k = 1 n ( n k ) ( − 1 ) k − 1 k . {\displaystyle {\begin{aligned}H_{n}&=\int _{0}^{1}{\frac {1-x^{n}}{1-x}}\,dx=\int _{0}^{1}{\frac {1-(1-u)^{n}}{u}}\,du\\[6pt]&=\int _{0}^{1}\left[\sum _{k=1}^{n}{\binom {n}{k}}(-u)^{k-1}\right]\,du=\sum _{k=1}^{n}{\binom {n}{k}}\int _{0}^{1}(-u)^{k-1}\,du\\[6pt]&=\sum _{k=1}^{n}{\binom {n}{k}}{\frac {(-1)^{k-1}}{k}}.\end{aligned}}} Graph demonstrating a connection between harmonic numbers and the natural logarithm . The harmonic number H n can be interpreted as a Riemann sum of the integral: ∫ 1 n + 1 d x x = ln ( n + 1 ) . {\displaystyle \int _{1}^{n+1}{\frac {dx}{x}}=\ln(n+1).} The n th harmonic number is about as large as the natural logarithm of n . The reason is that the sum is approximated by the integral
∫ 1 n 1 x d x , {\displaystyle \int _{1}^{n}{\frac {1}{x}}\,dx,} whose value is
ln n .
The values of the sequence H n − ln n decrease monotonically towards the limit
lim n → ∞ ( H n − ln n ) = γ , {\displaystyle \lim _{n\to \infty }\left(H_{n}-\ln n\right)=\gamma ,} where
γ ≈ 0.5772156649 is the
Euler–Mascheroni constant . The corresponding
asymptotic expansion is
H n ∼ ln n + γ + 1 2 n − ∑ k = 1 ∞ B 2 k 2 k n 2 k = ln n + γ + 1 2 n − 1 12 n 2 + 1 120 n 4 − ⋯ , {\displaystyle {\begin{aligned}H_{n}&\sim \ln {n}+\gamma +{\frac {1}{2n}}-\sum _{k=1}^{\infty }{\frac {B_{2k}}{2kn^{2k}}}\\&=\ln {n}+\gamma +{\frac {1}{2n}}-{\frac {1}{12n^{2}}}+{\frac {1}{120n^{4}}}-\cdots ,\end{aligned}}} where
B k are the
Bernoulli numbers .
Generating functions [ edit ] A generating function for the harmonic numbers is
∑ n = 1 ∞ z n H n = − ln ( 1 − z ) 1 − z , {\displaystyle \sum _{n=1}^{\infty }z^{n}H_{n}={\frac {-\ln(1-z)}{1-z}},} where ln(
z ) is the
natural logarithm . An exponential generating function is
∑ n = 1 ∞ z n n ! H n = e z ∑ k = 1 ∞ ( − 1 ) k − 1 k z k k ! = e z Ein ( z ) {\displaystyle \sum _{n=1}^{\infty }{\frac {z^{n}}{n!}}H_{n}=e^{z}\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}{\frac {z^{k}}{k!}}=e^{z}\operatorname {Ein} (z)} where Ein(
z ) is the entire
exponential integral . The exponential integral may also be expressed as
Ein ( z ) = E 1 ( z ) + γ + ln z = Γ ( 0 , z ) + γ + ln z {\displaystyle \operatorname {Ein} (z)=\mathrm {E} _{1}(z)+\gamma +\ln z=\Gamma (0,z)+\gamma +\ln z} where Γ(0,
z ) is the
incomplete gamma function .
Arithmetic properties [ edit ] The harmonic numbers have several interesting arithmetic properties. It is well-known that H n {\textstyle H_{n}} is an integer if and only if n = 1 {\textstyle n=1} , a result often attributed to Taeisinger.[5] Indeed, using 2-adic valuation , it is not difficult to prove that for n ≥ 2 {\textstyle n\geq 2} the numerator of H n {\textstyle H_{n}} is an odd number while the denominator of H n {\textstyle H_{n}} is an even number. More precisely,
H n = 1 2 ⌊ log 2 ( n ) ⌋ a n b n {\displaystyle H_{n}={\frac {1}{2^{\lfloor \log _{2}(n)\rfloor }}}{\frac {a_{n}}{b_{n}}}} with some odd integers
a n {\textstyle a_{n}} and
b n {\textstyle b_{n}} .
As a consequence of Wolstenholme's theorem , for any prime number p ≥ 5 {\displaystyle p\geq 5} the numerator of H p − 1 {\displaystyle H_{p-1}} is divisible by p 2 {\textstyle p^{2}} . Furthermore, Eisenstein[6] proved that for all odd prime number p {\textstyle p} it holds
H ( p − 1 ) / 2 ≡ − 2 q p ( 2 ) ( mod p ) {\displaystyle H_{(p-1)/2}\equiv -2q_{p}(2){\pmod {p}}} where
q p ( 2 ) = ( 2 p − 1 − 1 ) / p {\textstyle q_{p}(2)=(2^{p-1}-1)/p} is a
Fermat quotient , with the consequence that
p {\textstyle p} divides the numerator of
H ( p − 1 ) / 2 {\displaystyle H_{(p-1)/2}} if and only if
p {\textstyle p} is a
Wieferich prime .
In 1991, Eswarathasan and Levine[7] defined J p {\displaystyle J_{p}} as the set of all positive integers n {\displaystyle n} such that the numerator of H n {\displaystyle H_{n}} is divisible by a prime number p . {\displaystyle p.} They proved that
{ p − 1 , p 2 − p , p 2 − 1 } ⊆ J p {\displaystyle \{p-1,p^{2}-p,p^{2}-1\}\subseteq J_{p}} for all prime numbers
p ≥ 5 , {\displaystyle p\geq 5,} and they defined
harmonic primes to be the primes
p {\textstyle p} such that
J p {\displaystyle J_{p}} has exactly 3 elements.
Eswarathasan and Levine also conjectured that J p {\displaystyle J_{p}} is a finite set for all primes p , {\displaystyle p,} and that there are infinitely many harmonic primes. Boyd[8] verified that J p {\displaystyle J_{p}} is finite for all prime numbers up to p = 547 {\displaystyle p=547} except 83, 127, and 397; and he gave a heuristic suggesting that the density of the harmonic primes in the set of all primes should be 1 / e {\displaystyle 1/e} . Sanna[9] showed that J p {\displaystyle J_{p}} has zero asymptotic density , while Bing-Ling Wu and Yong-Gao Chen[10] proved that the number of elements of J p {\displaystyle J_{p}} not exceeding x {\displaystyle x} is at most 3 x 2 3 + 1 25 log p {\displaystyle 3x^{{\frac {2}{3}}+{\frac {1}{25\log p}}}} , for all x ≥ 1 {\displaystyle x\geq 1} .
Applications [ edit ] The harmonic numbers appear in several calculation formulas, such as the digamma function
ψ ( n ) = H n − 1 − γ . {\displaystyle \psi (n)=H_{n-1}-\gamma .} This relation is also frequently used to define the extension of the harmonic numbers to non-integer
n . The harmonic numbers are also frequently used to define
γ using the limit introduced earlier:
γ = lim n → ∞ ( H n − ln ( n ) ) , {\displaystyle \gamma =\lim _{n\rightarrow \infty }{\left(H_{n}-\ln(n)\right)},} although
γ = lim n → ∞ ( H n − ln ( n + 1 2 ) ) {\displaystyle \gamma =\lim _{n\to \infty }{\left(H_{n}-\ln \left(n+{\frac {1}{2}}\right)\right)}} converges more quickly.
In 2002, Jeffrey Lagarias proved[11] that the Riemann hypothesis is equivalent to the statement that
σ ( n ) ≤ H n + ( log H n ) e H n , {\displaystyle \sigma (n)\leq H_{n}+(\log H_{n})e^{H_{n}},} is true for every
integer n ≥ 1 with strict inequality if
n > 1; here
σ (n ) denotes the
sum of the divisors of
n .
The eigenvalues of the nonlocal problem
λ φ ( x ) = ∫ − 1 1 φ ( x ) − φ ( y ) | x − y | d y {\displaystyle \lambda \varphi (x)=\int _{-1}^{1}{\frac {\varphi (x)-\varphi (y)}{|x-y|}}\,dy} are given by
λ = 2 H n {\displaystyle \lambda =2H_{n}} , where by convention
H 0 = 0 {\displaystyle H_{0}=0} , and the corresponding eigenfunctions are given by the
Legendre polynomials φ ( x ) = P n ( x ) {\displaystyle \varphi (x)=P_{n}(x)} .
[12] Generalizations [ edit ] Generalized harmonic numbers [ edit ] The n th generalized harmonic number of order m is given by
H n , m = ∑ k = 1 n 1 k m . {\displaystyle H_{n,m}=\sum _{k=1}^{n}{\frac {1}{k^{m}}}.} (In some sources, this may also be denoted by H n ( m ) {\textstyle H_{n}^{(m)}} or H m ( n ) . {\textstyle H_{m}(n).} )
The special case m = 0 gives H n , 0 = n . {\displaystyle H_{n,0}=n.} The special case m = 1 reduces to the usual harmonic number:
H n , 1 = H n = ∑ k = 1 n 1 k . {\displaystyle H_{n,1}=H_{n}=\sum _{k=1}^{n}{\frac {1}{k}}.} The limit of H n , m {\textstyle H_{n,m}} as n → ∞ is finite if m > 1 , with the generalized harmonic number bounded by and converging to the Riemann zeta function
lim n → ∞ H n , m = ζ ( m ) . {\displaystyle \lim _{n\rightarrow \infty }H_{n,m}=\zeta (m).} The smallest natural number k such that kn does not divide the denominator of generalized harmonic number H (k , n ) nor the denominator of alternating generalized harmonic number H′ (k , n ) is, for n =1, 2, ... :
77, 20, 94556602, 42, 444, 20, 104, 42, 76, 20, 77, 110, 3504, 20, 903, 42, 1107, 20, 104, 42, 77, 20, 2948, 110, 136, 20, 76, 42, 903, 20, 77, 42, 268, 20, 7004, 110, 1752, 20, 19203, 42, 77, 20, 104, 42, 76, 20, 370, 110, 1107, 20, ... (sequence A128670 in the OEIS ) The related sum ∑ k = 1 n k m {\displaystyle \sum _{k=1}^{n}k^{m}} occurs in the study of Bernoulli numbers ; the harmonic numbers also appear in the study of Stirling numbers .
Some integrals of generalized harmonic numbers are
∫ 0 a H x , 2 d x = a π 2 6 − H a {\displaystyle \int _{0}^{a}H_{x,2}\,dx=a{\frac {\pi ^{2}}{6}}-H_{a}} and
∫ 0 a H x , 3 d x = a A − 1 2 H a , 2 , {\displaystyle \int _{0}^{a}H_{x,3}\,dx=aA-{\frac {1}{2}}H_{a,2},} where
A is
Apéry's constant ζ (3), and
∑ k = 1 n H k , m = ( n + 1 ) H n , m − H n , m − 1 for m ≥ 0. {\displaystyle \sum _{k=1}^{n}H_{k,m}=(n+1)H_{n,m}-H_{n,m-1}{\text{ for }}m\geq 0.} Every generalized harmonic number of order m can be written as a function of harmonic numbers of order m − 1 {\displaystyle m-1} using
H n , m = ∑ k = 1 n − 1 H k , m − 1 k ( k + 1 ) + H n , m − 1 n {\displaystyle H_{n,m}=\sum _{k=1}^{n-1}{\frac {H_{k,m-1}}{k(k+1)}}+{\frac {H_{n,m-1}}{n}}} for example:
H 4 , 3 = H 1 , 2 1 ⋅ 2 + H 2 , 2 2 ⋅ 3 + H 3 , 2 3 ⋅ 4 + H 4 , 2 4 {\displaystyle H_{4,3}={\frac {H_{1,2}}{1\cdot 2}}+{\frac {H_{2,2}}{2\cdot 3}}+{\frac {H_{3,2}}{3\cdot 4}}+{\frac {H_{4,2}}{4}}} A generating function for the generalized harmonic numbers is
∑ n = 1 ∞ z n H n , m = Li m ( z ) 1 − z , {\displaystyle \sum _{n=1}^{\infty }z^{n}H_{n,m}={\frac {\operatorname {Li} _{m}(z)}{1-z}},} where
Li m ( z ) {\displaystyle \operatorname {Li} _{m}(z)} is the
polylogarithm , and
|z | < 1 . The generating function given above for
m = 1 is a special case of this formula.
A fractional argument for generalized harmonic numbers can be introduced as follows:
For every p , q > 0 {\displaystyle p,q>0} integer, and m > 1 {\displaystyle m>1} integer or not, we have from polygamma functions:
H q / p , m = ζ ( m ) − p m ∑ k = 1 ∞ 1 ( q + p k ) m {\displaystyle H_{q/p,m}=\zeta (m)-p^{m}\sum _{k=1}^{\infty }{\frac {1}{(q+pk)^{m}}}} where
ζ ( m ) {\displaystyle \zeta (m)} is the
Riemann zeta function . The relevant recurrence relation is
H a , m = H a − 1 , m + 1 a m . {\displaystyle H_{a,m}=H_{a-1,m}+{\frac {1}{a^{m}}}.} Some special values are
H 1 4 , 2 = 16 − 5 6 π 2 − 8 G H 1 2 , 2 = 4 − π 2 3 H 3 4 , 2 = 16 9 − 5 6 π 2 + 8 G H 1 4 , 3 = 64 − π 3 − 27 ζ ( 3 ) H 1 2 , 3 = 8 − 6 ζ ( 3 ) H 3 4 , 3 = ( 4 3 ) 3 + π 3 − 27 ζ ( 3 ) {\displaystyle {\begin{aligned}H_{{\frac {1}{4}},2}&=16-{\tfrac {5}{6}}\pi ^{2}-8G\\H_{{\frac {1}{2}},2}&=4-{\frac {\pi ^{2}}{3}}\\H_{{\frac {3}{4}},2}&={\frac {16}{9}}-{\frac {5}{6}}\pi ^{2}+8G\\H_{{\frac {1}{4}},3}&=64-\pi ^{3}-27\zeta (3)\\H_{{\frac {1}{2}},3}&=8-6\zeta (3)\\H_{{\frac {3}{4}},3}&=\left({\frac {4}{3}}\right)^{3}+\pi ^{3}-27\zeta (3)\end{aligned}}} where
G is
Catalan's constant . In the special case that
p = 1 {\displaystyle p=1} , we get
H n , m = ζ ( m , 1 ) − ζ ( m , n + 1 ) , {\displaystyle H_{n,m}=\zeta (m,1)-\zeta (m,n+1),} where ζ ( m , n ) {\displaystyle \zeta (m,n)} is the Hurwitz zeta function . This relationship is used to calculate harmonic numbers numerically.
Multiplication formulas [ edit ] The multiplication theorem applies to harmonic numbers. Using polygamma functions, we obtain
H 2 x = 1 2 ( H x + H x − 1 2 ) + ln 2 H 3 x = 1 3 ( H x + H x − 1 3 + H x − 2 3 ) + ln 3 , {\displaystyle {\begin{aligned}H_{2x}&={\frac {1}{2}}\left(H_{x}+H_{x-{\frac {1}{2}}}\right)+\ln 2\\H_{3x}&={\frac {1}{3}}\left(H_{x}+H_{x-{\frac {1}{3}}}+H_{x-{\frac {2}{3}}}\right)+\ln 3,\end{aligned}}} or, more generally,
H n x = 1 n ( H x + H x − 1 n + H x − 2 n + ⋯ + H x − n − 1 n ) + ln n . {\displaystyle H_{nx}={\frac {1}{n}}\left(H_{x}+H_{x-{\frac {1}{n}}}+H_{x-{\frac {2}{n}}}+\cdots +H_{x-{\frac {n-1}{n}}}\right)+\ln n.} For generalized harmonic numbers, we have
H 2 x , 2 = 1 2 ( ζ ( 2 ) + 1 2 ( H x , 2 + H x − 1 2 , 2 ) ) H 3 x , 2 = 1 9 ( 6 ζ ( 2 ) + H x , 2 + H x − 1 3 , 2 + H x − 2 3 , 2 ) , {\displaystyle {\begin{aligned}H_{2x,2}&={\frac {1}{2}}\left(\zeta (2)+{\frac {1}{2}}\left(H_{x,2}+H_{x-{\frac {1}{2}},2}\right)\right)\\H_{3x,2}&={\frac {1}{9}}\left(6\zeta (2)+H_{x,2}+H_{x-{\frac {1}{3}},2}+H_{x-{\frac {2}{3}},2}\right),\end{aligned}}} where
ζ ( n ) {\displaystyle \zeta (n)} is the
Riemann zeta function .
Hyperharmonic numbers [ edit ] The next generalization was discussed by J. H. Conway and R. K. Guy in their 1995 book The Book of Numbers .[1] : 258 Let
H n ( 0 ) = 1 n . {\displaystyle H_{n}^{(0)}={\frac {1}{n}}.} Then the nth
hyperharmonic number of order
r (
r>0 ) is defined recursively as
H n ( r ) = ∑ k = 1 n H k ( r − 1 ) . {\displaystyle H_{n}^{(r)}=\sum _{k=1}^{n}H_{k}^{(r-1)}.} In particular,
H n ( 1 ) {\displaystyle H_{n}^{(1)}} is the ordinary harmonic number
H n {\displaystyle H_{n}} .
Roman Harmonic numbers [ edit ] The Roman Harmonic numbers ,[13] named after Steven Roman , were introduced by Daniel Loeb and Gian-Carlo Rota in the context of a generalization of umbral calculus with logarithms.[14] There are many possible definitions, but one of them, for n , k ≥ 0 {\displaystyle n,k\geq 0} , is
c n ( 0 ) = 1 , {\displaystyle c_{n}^{(0)}=1,} and
c n ( k + 1 ) = ∑ i = 1 n c i ( k ) i . {\displaystyle c_{n}^{(k+1)}=\sum _{i=1}^{n}{\frac {c_{i}^{(k)}}{i}}.} Of course,
c n ( 1 ) = H n . {\displaystyle c_{n}^{(1)}=H_{n}.} If n ≠ 0 {\displaystyle n\neq 0} , they satisfy
c n ( k + 1 ) − c n ( k ) n = c n − 1 ( k + 1 ) . {\displaystyle c_{n}^{(k+1)}-{\frac {c_{n}^{(k)}}{n}}=c_{n-1}^{(k+1)}.} Closed form formulas are
c n ( k ) = n ! ( − 1 ) k s ( − n , k ) , {\displaystyle c_{n}^{(k)}=n!(-1)^{k}s(-n,k),} where
s ( − n , k ) {\displaystyle s(-n,k)} is
Stirling numbers of the first kind generalized to negative first argument, and
c n ( k ) = ∑ j = 1 n ( n j ) ( − 1 ) j − 1 j k , {\displaystyle c_{n}^{(k)}=\sum _{j=1}^{n}{\binom {n}{j}}{\frac {(-1)^{j-1}}{j^{k}}},} which was found by
Donald Knuth .
In fact, these numbers were defined in a more general manner using Roman numbers and Roman factorials , that include negative values for n {\displaystyle n} . This generalization was useful in their study to define Harmonic logarithms .
Harmonic numbers for real and complex values [ edit ] The formulae given above,
H x = ∫ 0 1 1 − t x 1 − t d t = ∑ k = 1 ∞ ( x k ) ( − 1 ) k − 1 k {\displaystyle H_{x}=\int _{0}^{1}{\frac {1-t^{x}}{1-t}}\,dt=\sum _{k=1}^{\infty }{x \choose k}{\frac {(-1)^{k-1}}{k}}} are an integral and a series representation for a function that interpolates the harmonic numbers and, via
analytic continuation , extends the definition to the complex plane other than the negative integers
x . The interpolating function is in fact closely related to the
digamma function H x = ψ ( x + 1 ) + γ , {\displaystyle H_{x}=\psi (x+1)+\gamma ,} where
ψ (x ) is the digamma function, and
γ is the
Euler–Mascheroni constant . The integration process may be repeated to obtain
H x , 2 = ∑ k = 1 ∞ ( − 1 ) k − 1 k ( x k ) H k . {\displaystyle H_{x,2}=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}{x \choose k}H_{k}.} The Taylor series for the harmonic numbers is
H x = ∑ k = 2 ∞ ( − 1 ) k ζ ( k ) x k − 1 for | x | < 1 {\displaystyle H_{x}=\sum _{k=2}^{\infty }(-1)^{k}\zeta (k)\;x^{k-1}\quad {\text{ for }}|x|<1} which comes from the Taylor series for the digamma function (
ζ {\displaystyle \zeta } is the
Riemann zeta function ).
Alternative, asymptotic formulation [ edit ] When seeking to approximate H x for a complex number x , it is effective to first compute H m for some large integer m . Use that as an approximation for the value of H m +x . Then use the recursion relation H n = H n −1 + 1/n backwards m times, to unwind it to an approximation for H x . Furthermore, this approximation is exact in the limit as m goes to infinity.
Specifically, for a fixed integer n , it is the case that
lim m → ∞ [ H m + n − H m ] = 0. {\displaystyle \lim _{m\rightarrow \infty }\left[H_{m+n}-H_{m}\right]=0.} If n is not an integer then it is not possible to say whether this equation is true because we have not yet (in this section) defined harmonic numbers for non-integers. However, we do get a unique extension of the harmonic numbers to the non-integers by insisting that this equation continue to hold when the arbitrary integer n is replaced by an arbitrary complex number x ,
lim m → ∞ [ H m + x − H m ] = 0 . {\displaystyle \lim _{m\rightarrow \infty }\left[H_{m+x}-H_{m}\right]=0\,.} Swapping the order of the two sides of this equation and then subtracting them from
H x gives
H x = lim m → ∞ [ H m − ( H m + x − H x ) ] = lim m → ∞ [ ( ∑ k = 1 m 1 k ) − ( ∑ k = 1 m 1 x + k ) ] = lim m → ∞ ∑ k = 1 m ( 1 k − 1 x + k ) = x ∑ k = 1 ∞ 1 k ( x + k ) . {\displaystyle {\begin{aligned}H_{x}&=\lim _{m\rightarrow \infty }\left[H_{m}-(H_{m+x}-H_{x})\right]\\[6pt]&=\lim _{m\rightarrow \infty }\left[\left(\sum _{k=1}^{m}{\frac {1}{k}}\right)-\left(\sum _{k=1}^{m}{\frac {1}{x+k}}\right)\right]\\[6pt]&=\lim _{m\rightarrow \infty }\sum _{k=1}^{m}\left({\frac {1}{k}}-{\frac {1}{x+k}}\right)=x\sum _{k=1}^{\infty }{\frac {1}{k(x+k)}}\,.\end{aligned}}} This infinite series converges for all complex numbers x except the negative integers, which fail because trying to use the recursion relation H n = H n −1 + 1/n backwards through the value n = 0 involves a division by zero. By this construction, the function that defines the harmonic number for complex values is the unique function that simultaneously satisfies (1) H 0 = 0 , (2) H x = H x −1 + 1/x for all complex numbers x except the non-positive integers, and (3) limm →+∞ (H m +x − H m ) = 0 for all complex values x .
This last formula can be used to show that
∫ 0 1 H x d x = γ , {\displaystyle \int _{0}^{1}H_{x}\,dx=\gamma ,} where
γ is the
Euler–Mascheroni constant or, more generally, for every
n we have:
∫ 0 n H x d x = n γ + ln ( n ! ) . {\displaystyle \int _{0}^{n}H_{x}\,dx=n\gamma +\ln(n!).} Special values for fractional arguments [ edit ] There are the following special analytic values for fractional arguments between 0 and 1, given by the integral
H α = ∫ 0 1 1 − x α 1 − x d x . {\displaystyle H_{\alpha }=\int _{0}^{1}{\frac {1-x^{\alpha }}{1-x}}\,dx\,.} More values may be generated from the recurrence relation
H α = H α − 1 + 1 α , {\displaystyle H_{\alpha }=H_{\alpha -1}+{\frac {1}{\alpha }}\,,} or from the reflection relation
H 1 − α − H α = π cot ( π α ) − 1 α + 1 1 − α . {\displaystyle H_{1-\alpha }-H_{\alpha }=\pi \cot {(\pi \alpha )}-{\frac {1}{\alpha }}+{\frac {1}{1-\alpha }}\,.} For example:
H 1 2 = 2 − 2 ln 2 H 1 3 = 3 − π 2 3 − 3 2 ln 3 H 2 3 = 3 2 + π 2 3 − 3 2 ln 3 H 1 4 = 4 − π 2 − 3 ln 2 H 3 4 = 4 3 + π 2 − 3 ln 2 H 1 6 = 6 − 3 2 π − 2 ln 2 − 3 2 ln 3 H 1 8 = 8 − 1 + 2 2 π − 4 ln 2 − 1 2 ( ln ( 2 + 2 ) − ln ( 2 − 2 ) ) H 1 12 = 12 − ( 1 + 3 2 ) π − 3 ln 2 − 3 2 ln 3 + 3 ln ( 2 − 3 ) {\displaystyle {\begin{aligned}H_{\frac {1}{2}}&=2-2\ln 2\\H_{\frac {1}{3}}&=3-{\frac {\pi }{2{\sqrt {3}}}}-{\frac {3}{2}}\ln 3\\H_{\frac {2}{3}}&={\frac {3}{2}}+{\frac {\pi }{2{\sqrt {3}}}}-{\frac {3}{2}}\ln 3\\H_{\frac {1}{4}}&=4-{\frac {\pi }{2}}-3\ln 2\\H_{\frac {3}{4}}&={\frac {4}{3}}+{\frac {\pi }{2}}-3\ln 2\\H_{\frac {1}{6}}&=6-{\frac {\sqrt {3}}{2}}\pi -2\ln 2-{\frac {3}{2}}\ln 3\\H_{\frac {1}{8}}&=8-{\frac {1+{\sqrt {2}}}{2}}\pi -4\ln {2}-{\frac {1}{\sqrt {2}}}\left(\ln \left(2+{\sqrt {2}}\right)-\ln \left(2-{\sqrt {2}}\right)\right)\\H_{\frac {1}{12}}&=12-\left(1+{\frac {\sqrt {3}}{2}}\right)\pi -3\ln {2}-{\frac {3}{2}}\ln {3}+{\sqrt {3}}\ln \left(2-{\sqrt {3}}\right)\end{aligned}}} Which are computed via Gauss's digamma theorem , which essentially states that for positive integers p and q with p < q
H p q = q p + 2 ∑ k = 1 ⌊ q − 1 2 ⌋ cos ( 2 π p k q ) ln ( sin ( π k q ) ) − π 2 cot ( π p q ) − ln ( 2 q ) {\displaystyle H_{\frac {p}{q}}={\frac {q}{p}}+2\sum _{k=1}^{\lfloor {\frac {q-1}{2}}\rfloor }\cos \left({\frac {2\pi pk}{q}}\right)\ln \left({\sin \left({\frac {\pi k}{q}}\right)}\right)-{\frac {\pi }{2}}\cot \left({\frac {\pi p}{q}}\right)-\ln \left(2q\right)} Relation to the Riemann zeta function [ edit ] Some derivatives of fractional harmonic numbers are given by
d n H x d x n = ( − 1 ) n + 1 n ! [ ζ ( n + 1 ) − H x , n + 1 ] d n H x , 2 d x n = ( − 1 ) n + 1 ( n + 1 ) ! [ ζ ( n + 2 ) − H x , n + 2 ] d n H x , 3 d x n = ( − 1 ) n + 1 1 2 ( n + 2 ) ! [ ζ ( n + 3 ) − H x , n + 3 ] . {\displaystyle {\begin{aligned}{\frac {d^{n}H_{x}}{dx^{n}}}&=(-1)^{n+1}n!\left[\zeta (n+1)-H_{x,n+1}\right]\\[6pt]{\frac {d^{n}H_{x,2}}{dx^{n}}}&=(-1)^{n+1}(n+1)!\left[\zeta (n+2)-H_{x,n+2}\right]\\[6pt]{\frac {d^{n}H_{x,3}}{dx^{n}}}&=(-1)^{n+1}{\frac {1}{2}}(n+2)!\left[\zeta (n+3)-H_{x,n+3}\right].\end{aligned}}} And using Maclaurin series , we have for x < 1 that
H x = ∑ n = 1 ∞ ( − 1 ) n + 1 x n ζ ( n + 1 ) H x , 2 = ∑ n = 1 ∞ ( − 1 ) n + 1 ( n + 1 ) x n ζ ( n + 2 ) H x , 3 = 1 2 ∑ n = 1 ∞ ( − 1 ) n + 1 ( n + 1 ) ( n + 2 ) x n ζ ( n + 3 ) . {\displaystyle {\begin{aligned}H_{x}&=\sum _{n=1}^{\infty }(-1)^{n+1}x^{n}\zeta (n+1)\\[5pt]H_{x,2}&=\sum _{n=1}^{\infty }(-1)^{n+1}(n+1)x^{n}\zeta (n+2)\\[5pt]H_{x,3}&={\frac {1}{2}}\sum _{n=1}^{\infty }(-1)^{n+1}(n+1)(n+2)x^{n}\zeta (n+3).\end{aligned}}} For fractional arguments between 0 and 1 and for a > 1,
H 1 / a = 1 a ( ζ ( 2 ) − 1 a ζ ( 3 ) + 1 a 2 ζ ( 4 ) − 1 a 3 ζ ( 5 ) + ⋯ ) H 1 / a , 2 = 1 a ( 2 ζ ( 3 ) − 3 a ζ ( 4 ) + 4 a 2 ζ ( 5 ) − 5 a 3 ζ ( 6 ) + ⋯ ) H 1 / a , 3 = 1 2 a ( 2 ⋅ 3 ζ ( 4 ) − 3 ⋅ 4 a ζ ( 5 ) + 4 ⋅ 5 a 2 ζ ( 6 ) − 5 ⋅ 6 a 3 ζ ( 7 ) + ⋯ ) . {\displaystyle {\begin{aligned}H_{1/a}&={\frac {1}{a}}\left(\zeta (2)-{\frac {1}{a}}\zeta (3)+{\frac {1}{a^{2}}}\zeta (4)-{\frac {1}{a^{3}}}\zeta (5)+\cdots \right)\\[6pt]H_{1/a,\,2}&={\frac {1}{a}}\left(2\zeta (3)-{\frac {3}{a}}\zeta (4)+{\frac {4}{a^{2}}}\zeta (5)-{\frac {5}{a^{3}}}\zeta (6)+\cdots \right)\\[6pt]H_{1/a,\,3}&={\frac {1}{2a}}\left(2\cdot 3\zeta (4)-{\frac {3\cdot 4}{a}}\zeta (5)+{\frac {4\cdot 5}{a^{2}}}\zeta (6)-{\frac {5\cdot 6}{a^{3}}}\zeta (7)+\cdots \right).\end{aligned}}} See also [ edit ] ^ a b John H., Conway; Richard K., Guy (1995). The book of numbers . Copernicus. ^ Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994). Concrete Mathematics . Addison-Wesley. ^ Sondow, Jonathan and Weisstein, Eric W. "Harmonic Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HarmonicNumber.html ^ Sandifer, C. Edward (2007), How Euler Did It , MAA Spectrum, Mathematical Association of America, p. 206, ISBN 9780883855638 . ^ Weisstein, Eric W. (2003). CRC Concise Encyclopedia of Mathematics . Boca Raton, FL: Chapman & Hall/CRC. p. 3115. ISBN 978-1-58488-347-0 . ^ Eisenstein, Ferdinand Gotthold Max (1850). "Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen ahhängen und durch gewisse lineare Funktional-Gleichungen definirt werden". Berichte Königl. Preuβ. Akad. Wiss. Berlin . 15 : 36–42. ^ Eswarathasan, Arulappah; Levine, Eugene (1991). "p-integral harmonic sums" . Discrete Mathematics . 91 (3): 249–257. doi :10.1016/0012-365X(90)90234-9 . ^ Boyd, David W. (1994). "A p-adic study of the partial sums of the harmonic series" . Experimental Mathematics . 3 (4): 287–302. CiteSeerX 10.1.1.56.7026 . doi :10.1080/10586458.1994.10504298 . ^ Sanna, Carlo (2016). "On the p-adic valuation of harmonic numbers" (PDF) . Journal of Number Theory . 166 : 41–46. doi :10.1016/j.jnt.2016.02.020 . hdl :2318/1622121 . ^ Chen, Yong-Gao; Wu, Bing-Ling (2017). "On certain properties of harmonic numbers". Journal of Number Theory . 175 : 66–86. doi :10.1016/j.jnt.2016.11.027 . ^ Jeffrey Lagarias (2002). "An Elementary Problem Equivalent to the Riemann Hypothesis". Amer. Math. Monthly . 109 (6): 534–543. arXiv :math.NT/0008177 . doi :10.2307/2695443 . JSTOR 2695443 . ^ E.O. Tuck (1964). "Some methods for flows past blunt slender bodies". J. Fluid Mech . 18 (4): 619–635. Bibcode :1964JFM....18..619T . doi :10.1017/S0022112064000453 . S2CID 123120978 . ^ Sesma, J. (2017). "The Roman harmonic numbers revisited" . Journal of Number Theory . 180 : 544–565. arXiv :1702.03718 . doi :10.1016/j.jnt.2017.05.009 . ISSN 0022-314X . ^ Loeb, Daniel E; Rota, Gian-Carlo (1989). "Formal power series of logarithmic type" . Advances in Mathematics . 75 (1): 1–118. doi :10.1016/0001-8708(89)90079-0 . ISSN 0001-8708 . References [ edit ] External links [ edit ] This article incorporates material from Harmonic number on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License .